Class 12 Maths - KARNATAKA
Relations and Functions
The chapter 'Relations and Functions' in Class 12 Mathematics builds upon foundational set theory from Class 11, exploring types of relations such as reflexive, symmetric, transitive, and equivalence relations. It also covers types of functions including one-one (injective), onto (surjective), and bijective functions, alongside the concept of invertible functions and binary operations. This chapter is vital for Karnataka (KSEEB) board exams as it forms the basis of advanced calculus and algebra. Students regularly encounter 1-mark, 2-mark, and 5-mark conceptual problems requiring rigorous proof-writing based on definitions rather than just numerical calculations.
Start Learning FreeKey Concepts
Reflexive Relation
A relation R on a set A is reflexive if every element 'a' in A is related to itself, meaning (a, a) belongs to R for all a in A.
Symmetric Relation
A relation R on set A is symmetric if whenever (a, b) belongs to R, then (b, a) must also belong to R for all a, b in A.
Transitive Relation
A relation R on set A is transitive if whenever (a, b) and (b, c) belong to R, then (a, c) must also belong to R for all a, b, c in A.
Equivalence Relation
A relation is called an equivalence relation if it is simultaneously reflexive, symmetric, and transitive.
One-One Function (Injective)
A function f: A -> B is one-one if distinct elements of A have distinct images in B, meaning f(x1) = f(x2) implies x1 = x2.
Onto Function (Surjective)
A function f: A -> B is onto if every element in the co-domain B has at least one pre-image in the domain A, meaning Range = Co-domain.
Important Formulas
Board Exam Info
In the Karnataka (KSEEB) Class 12 Mathematics board examination, this chapter typically carries around 7 to 10 marks. Questions usually include one 1-mark MCQ or fill-in-the-blank, one 2-mark or 3-mark question, and a compulsory 5-mark long-answer question frequently focused on proving whether a given relation is an equivalence relation or checking the invertibility of a function.
Frequently Asked Questions
How do I prove a relation is an equivalence relation?
You must separately prove three properties: Reflexive (show (a,a) in R), Symmetric (show if (a,b) in R then (b,a) in R), and Transitive (show if (a,b) and (b,c) in R then (a,c) in R), using general variables, not just numbers.
What is the easiest way to prove a function is one-one?
Assume that f(x1) = f(x2) for any two elements x1 and x2 in the domain, and algebraically solve it to finally show that x1 = x2.
Can a function be onto if the co-domain is larger than the range?
No, a function is onto only if the range is strictly equal to the co-domain. If there is any element in the co-domain with no pre-image, the function is not onto.
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